Firoozbakht's conjecture

It is named after the Iranian mathematician Farideh Firoozbakht who stated it in 1982.

is the nth prime) is a strictly decreasing function of n, i.e., Equivalently: see OEIS: A182134, OEIS: A246782.

By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×1012.

[2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 264 ≈ 1.84×1019.

[3][4][5] If the conjecture were true, then the prime gap function

This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures.

[4] It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz[8][9][10] and of Maier[11][12] which suggest that occurs infinitely often for any

Three related conjectures (see the comments of OEIS: A182514) are variants of Firoozbakht's.

Forgues notes that Firoozbakht's can be written where the right hand side is the well-known expression which reaches Euler's number in the limit

, suggesting the slightly weaker conjecture Nicholson and Farhadian[13][14] give two stronger versions of Firoozbakht's conjecture which can be summarized as: where the right-hand inequality is Firoozbakht's, the middle is Nicholson's (since

; see Prime number theorem § Non-asymptotic bounds on the prime-counting function), and the left-hand inequality is Farhadian's (since